Fredholm determinant

Summary

The Fredholm determinant of a continuous kernel $K$ on $[a, b]$ is the entire function $D(\lambda) = \sum_{n \ge 0} \frac{(-\lambda)^n}{n!} \int \cdots \int \det[K(x_i, x_j)] \, dx_1 \cdots dx_n$. It plays for the integral operator $\varphi \mapsto \int K(\cdot, y) \varphi(y) \, dy$ the role that $\det(I - \lambda A)$ plays for a square matrix $A$. Its zeros are the characteristic values of the kernel. Together with the Fredholm minors it gives an explicit formula for the solution of the Fredholm equation of the second kind. When the integral operator $T$ of $K$ is a trace class operator, $D(\lambda)$ equals the operator determinant $\det(I - \lambda T)$.